In our scheme, we only consider the atoms in the dressed states
\(|\chi _1\rangle \)  and \(|\chi _4\rangle \) . Thus, we have to
adiabatically eliminate the populations of the dressed states
\(|\chi _2\rangle \)  and \(|\chi _3\rangle \)  and to avoid the atoms
decaying into these two dressed states. This can be realized in the
steps. First, we start with the atoms in the atomic state
\(|1\rangle \)  and \(\Omega _1=0\) , \(\Omega _3=0\)  with \(\Omega _2\)  finite,
then slowly turn \(\Omega _1\) , we will end up with the atoms in the
dressed state \(|\chi _1\rangle \)  [1]}. During this process,
the variation of \(\Omega _1\)  is slow enough to satisfy the adiabatic
condition \(|\langle \chi _j|\partial /\partial t|\chi _1\rangle |\ll |E_j-E_1|/\hbar =\Omega \)  with \(j=2,3\)  [2]}, [3]}. In the
second step, we adiabatically turn the Rabi frequency \(\Omega _3\)  on,
we will end up with atoms in the dressed states \(|\chi _1\rangle \)  and
\(|\chi _4\rangle \) . To avoid the atoms decaying into the dressed
states \(|\chi _2\rangle \)  and \(|\chi _3\rangle \) , the adiabatic
conditions \(\frac{1}{2m}|\tilde{\bf A}_{j1}|^2=\frac{\hbar ^2}{2m}q^2\sin ^2\theta \cos ^2\theta \ll |E_{j}-E_{1}|=\hbar \Omega \)  and
\(|\tilde{N}_{j4}|=\frac{\sqrt{2}}{2}\hbar \Omega _3\sin \theta \ll |E_{j}-E_4|=\hbar \Omega \)  for \(j=2,3\)  are satisfied. This is to say,
the off-diagonal elements of the Hamiltonian are small enough to
avoid the atoms decaying into the dressed states \(|\chi _2\rangle \) 
and \(|\chi _3\rangle \) .
